Some Formal Results on Positivity, Stability, and Endemic Steady-State Attainability Based on Linear Algebraic Tools for a Class of Epidemic Models with Eventual Incommensurate Delays

A formal description of typical compartmental epidemic models obtained is presented by splitting the state into an infective substate, or infective compartment, and a noninfective substate, or noninfective compartment. A general formal study to obtain the reproduction number and discuss the positivity and stability properties of equilibrium points is proposed and formally discussed. Such a study unifies previous related research and it is based on linear algebraic tools to investigate the positivity and the stability of the linearized dynamics around the disease-free and endemic equilibrium points. To this end, the complete state vector is split into the dynamically coupled infective and noninfective compartments each one containing the corresponding state components. The study is then extended to the case of commensurate internal delays when all the delays are integer multiples of a base delay. Two auxiliary delay-free systems are defined related to the linearization processes around the equilibrium points which correspond to the zero delay, i.e., delay-free, and infinity delay cases. Those auxiliary systems are used to formulate stability and positivity properties independently of the delay sizes. Some examples are discussed to the light of the developed formal study.The authors are grateful to the Spanish Government for Grants DPI2015-64766-R, RTI2018-094336-B-I00, and DPI2016-77271-R (MINECO/FEDER, UE)

Supply Chain Operations Management in Pandemics: A State-of-the-Art Review Inspired by COVID-19

Pandemics cause chaotic situations in supply chains (SC) around the globe, which can lead towards survivability challenges. The ongoing COVID-19 pandemic is an unprecedented humanitarian crisis that has severely affected global business dynamics. Similar vulnerabilities have been caused by other outbreaks in the past. In these terms, prevention strategies against propagating disruptions require vigilant goal conceptualization and roadmaps. In this respect, there is a need to explore supply chain operation management strategies to overcome the challenges that emerge due to COVID-19-like situations. Therefore, this review is aimed at exploring such challenges and developing strategies for sustainability, and viability perspectives for SCs, through a structured literature review (SLR) approach. Moreover, this study investigated the impacts of previous epidemic outbreaks on SCs, to identify the research objectives, methodological approaches, and implications for SCs. The study also explored the impacts of epidemic outbreaks on the business environment, in terms of effective resource allocation, supply and demand disruptions, and transportation network optimization, through operations management techniques. Furthermore, this article structured a framework that emphasizes the integration of Industry 4.0 technologies, resilience strategies, and sustainability to overcome SC challenges during pandemics. Finally, future research avenues were identified by including a research agenda for experts and practitioners to develop new pathways to get out of the crisis.</jats:p